GEOMETRY / TOPOLOGY CONFERENCE
May 25 - May 30 (2015)
List of invited speakers/participants
| C. Manolescu||
|| E. Murphy ||
|| D. Ruberman |
| F. Lin ||
|| M. Kreck ||
|| D. Auckly |
| A. Wand ||
|| A. Stipsicz ||
|| G. Dimitroglou Rizell|
| P. Ghiggini ||
|| J. Nelson ||
|| B. Chantraine |
| I. Petkova ||
|| S. Galkin ||
|| C. Blanchet |
| B. Ozbagci ||
|| T. Dereli ||
|| S. Finashin |
| M. Bhupal ||
|| M. Kalafat ||
|| I. Unal |
|| S. Uguz ||
Scientific Committee : D. Auroux, Y. Eliashberg, I. Itenberg, G. Mikhalkin, S. Akbulut
Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, Ö. Kişisel, Y. Ozan, T. Etgü, S. Salur
Supporting Organizations: NSF (National Science Foundation)
TMD (Turkish Mathematical Society).
The participants of 22nd Gökova Geometry - Topology Conference
List of Talks
| Ciprian Manolescu ||
||Mini course: The triangulation conjecture and related results
The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. The goal of this mini-course is to explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology. Time permitting, I will also discuss other related developments: an application of Pin(2)-equivariant Seiberg-Witten Floer K-theory (to the intersection forms of spin four-manifolds with boundary), and a construction (joint with Kristen Hendricks) of an involutive variant of Heegaard Floer homology - the analog of Z4-equivariant Seiberg-Witten Floer homology.
| Emmy Murphy ||
|| Mini course: Geometry of contact structures and overtwistedness in high
We discuss a number of geometric pictures of contact
manifolds, focusing on interactions of these pictures with the recent
concept of overtwistedness. The main topics we will cover are:
Legendrian manifolds and looseness, Weinstein cobordisms and contact
surgery, the plastikstufe and fillability, and open book
decompositions. Many of these ideas have been around for a long time.
However the relationships between these pictures have only begun to
take shape, spurred on by the recent discovery of overtwistedness.
| Daniel Ruberman ||
|| Absolutely exotic 4-manifolds |
We show the existence of exotic smooth structures on contractible 4-manifolds. These structures are absolute, in the sense that they do not depend on a specific marking of the boundary. This is in contrast to the phenomenon of corks, which are exotic relative to an automorphism of their boundaries. The technique is to modify a relatively exotic manifold to give an exotic one for which we have a good understanding of the automorphism group of the boundary. This is joint work with Selman Akbulut.
| Jo Nelson ||
|| An integral lift of contact homology |
Cylindrical contact homology is arguably one of the more notorious Floer-theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations has tarnished its claim to being a well-defined contact invariant. However, recent work of Hutchings and Nelson has managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of non-equivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to SH+ under suitable assumptions, though it does not require a filling of the contact manifold. By making use of family Floer theory we obtain an S1-equivariant theory defined over Z coefficients, which when tensored with Q yields cylindrical contact homology, now with the guarantee of well-definedness and invariance.
| Ina Petkova ||
|| Combinatorial tangle Floer homology |
In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss the construction of this combinatorial invariant for tangles in S3, D3, and I x S2. The special case of S3 gives back a stabilized version of knot Floer homology. If time permits, I will discuss how tangle Floer homology enhances the structure of knot Floer homology, focusing on its gluing properties, its similarities with other theories, and its relation to quantum topology.
| Matthias Kreck ||
|| Manifolds which are like the associative Grassmannian (joint work with D. Crowley and D. Salamon) |
The associative Grassmannian is the homogeneous space G2/SO(4), where G2 is the exceptional Lie group of dimension 14. Akbulut and Kalafat have determined the cohomology ring which gives a rational homology quaternionic projective plane. Eells and Kuiper have written a paper "Manifolds which are like the projective plane" and investigated the diffeomorphism type of simply connected manifolds which have the cohomology ring of the projective planes. We do the same with the associative Grassmannian instead of the projective plane.
| Andras Stipsicz ||
|| Knot Floer homologies |
Knot Floer homology (introduced by Ozsvath-Szabo and independently by
Rasmussen) is a powerful tool for studying knots and links in the 3-sphere. In
particular, it gives rise to a numerical invariant, which provides a
nontrivial lower bound on the 4-dimensional genus of the knot. By deforming
the definition of knot Floer homology by a real number t from [0,2], we define
a family of homologies, and derive a family of numerical invariants with
similar properties. The resulting invariants provide a family of
homomorphisms on the concordance group. One of these homomorphisms can be
used to estimate the unoriented 4-dimensional genus of the knot. We will
review the basic constructions for knot Floer homology and the deformed
theories and discuss some of the applications. This is joint work with
P. Ozsvath and Z. Szabo.
| Georgios Dimitroglou Rizell ||
|| The classification of Lagrangian tori in a four-dimensional symplectic vectorspace |
We prove that there are exactly two monotone Lagrangian tori in a four-dimensional symplectic vectorspace up to Hamiltonian isotopy and rescaling: the Clifford torus and the Chekanov torus. Moreover, all non-monotone tori are shown to be Hamiltonian isotopic to product tori. The strategy is, first, finding a singular symplectic conic linking the torus appropriately and, second, applying a classification result for homotopically non-trivial Lagrangian tori inside the cotangent bundle of a two-torus. The latter result follows using methods due to Ivrii.
| Andy Wand ||
|| Tightness and open book decompositions |
We will discuss a characterization of tightness of a contact 3-manifold in terms of supporting open book decompositions, some applications, and generalizations.
| Francesco Lin ||
|| The surgery exact triangle in Pin(2)-monopole Floer homology |
In this talk we discuss an approach to Manolescu's Pin(2)-equivariant Seiberg-Witten Floer homology
from Kronheimer and Mrowka's Morse theoretic point of view, and discuss the relationship between
the invariants associated to surgeries on a given knot.
| Paolo Ghiggini ||
|| Tight contact structures on connected sums which are not contact sonnected sum |
It is well known that, in dimension three, every tight contact structure on a connected sum
is a contact connected sum. I will show that the same statement is not true in dimension five.
This is a joint work with Klaus Niederkrüger and Chris Wendl.
| Christian Blanchet ||
|| Non semisimple TQFTs from nilpotent representations of quantum sl(2) |
A new family of quantum invariants based on nilpotent representions of quantum sl(2)
at a root of unity have been constructed by Costantino-Geer-Patureau. We show that the
new quantum invariants have graded TQFT extensions. We will consider the specific case of
root of unity of order 4 where CGP invariants recover Reidemeister torsion with canonical
normalisation. In the general case we will describe the arising representations of mapping class groups.
(Joint work with François Costantino, Nathan Geer and Bertrand Patureau)